[r-t] What's the meaning of a method having aparticularfalsecourse head

[Richard Smith]

Don Morrison wrote:

> On Apr 23, 2005, at 9:13 AM, Graham John wrote:
>
> > 1. A Handbook of Composition by John Leary (CC pub 1993)
> >
> > Appendix 1 pg 51.
> >
> > "Group A is not shown. It contains the remaining in-course FCH, which is
> > 23456, and represents the trivial case of the plain course being false
> > against itself."

I'm inclined to take the word "trivial" as support for my
position.

> Perhaps it's simply a reflection of my own bias, but I'm still puzzled
> about why those preferring the "A falseness means the trivial case all
> methods have of the same course being rung twice being false against
> itself" interpretation do prefer it.

Because, to my mind, the most logical definition of
falseness (which I've given below) has 23456 as a FCH of all
methods.  As I tried to say in a previous email, I put more
value on the scheme being consistent in its treatment of A
falseness, than of it assigning a useful meaning to A
falseness.  I accept that by defining A falseness to mean
falseness within the plain course, you might end up with a
more useful definition of A falseness than mine.  However, I
think my definition is much more consistent.  This is why I
previously labelled the two persectives as pragmatic and
purist, respectively.

> It was at some point in this
> discussion described as an "identity" but that can't really be the case
> here, can it? The false course head "groups" do not form a mathematical
> group, and I'm unaware of anyone ever having defined an operation on
> them for which there would be an identity.

They don't form a group, but the set of all combinations of
falseness "groups" form a monoid -- a mathematical structure
similar to a group, but without the requirement for elements
to have inverses.  The A falseness "group" forms the
identity.  (This isn't a particularly insightful statement,
but it does justify using the term "identity" to describe A
falseness.)

> I suppose something could be
> defined just out of perversity, but is there any utility in having an
> identity for the elements of the partitioning of FCHs?

It's also the partition containing the identity FCH -- and
as the set of FCHs (as opposed to falseness "groups") does
actually form a group, this is a meaningful statement.

> I'm intrigued, too, that no one has yet really offered a definition of
> what it means for a method to have a particular FCH. [...]
> I would have thought it would be something like
>
> i) A method is said to have FCH x if and only if two rows, a and b,
> occurring at different points in its plain course are related such that
> x permuted by a = b or x permuted by b = a.
>
> Richard's interpretation would, I think, be
>
> ii) A method is said to have FCH x if and only if two rows, a and b,
> possibly different ones or possibly the same row, are related such that
> x permuted by a = b or x permuted by b = a.

I wouldn't personally take either of these as the
definition, though my prefered definition would imply and be
implied by definition (ii).  Both of these look to me more
like simple results one would prove from a definition than a
definition itself.

Here's my definition:

  Let M be the set of a rows in the plain course of a
  method, and let rM denote the set of rows in the course
  with course head row, r:

    rM = { ra : a in M }.

  Define the set, F, of false course heads of M to be:

    F = { f : fM intersection M != {} }.



This definition seems much better as it is explicit as to
what it means -- if course f and the plain course have a row
in common, then they are false, and f is refered to a as a
FCH.  It is fairly straightforward to derive your definition
(ii)  from this.

This definition also fits better with a natural definition
of "false":  two sets of rows, A and B are mutually
false if they have a non-empty intersection.  Rephrasing the
definition to use this terminology, we can say that f is a
FCH of M if and only if M and fM are mutually false.


> Anyway, can anyone in the "Richard's interpretation" camp perhaps
> explain a little more *why* you favour the interpretation you do?

I hope the above goes some way towards doing that.

Richard